Question

Determine the inverse of the function f(x) = 3(x - 4)^2 + 5.

Answer 1

Answer:(x) as given has no inverse. We see that, for instance, f(1) = f(7) = 32; that is, two different values of x give the same value of f(x), so f is not one-to-one.We can however restrict the domain over which f is defined to extract an invertible function. One such choice would be to restrict to the interval x ≥ 4, which I'll demonstrate below.Recall the definition of function inverse:f(f^(-1)(x)) = xThen for this function, we have3 (f^(-1)(x) - 4)² + 5 = x3 (f^(-1)(x) - 4)² = x - 5(f^(-1)(x) - 4)² = (x - 5)/3√((f^(-1)(x) - 4)²)= √((x - 5)/3)For all x, √(x²) = |x|. Then √((f^(-1)(x) - 4)²) = |f^(-1)(x) - 4|, but by restricting x ≥ 4, or x - 4 ≥ 0, we have the condition that f^(-1)(x) - 4 ≥ 0, and so by definition (of the absolute value function) the absolute value reduces to the positive case. In short, we end up withf^(-1)(x) - 4 = √((x - 5)/3)and hence the inversef^(-1)(x) = √((x - 5)/3) + 4

Step-by-step explanation:

Answer 2

f(x) as given has no inverse. We see that, for instance, f(1) = f(7) = 32; that is, two different values of x give the same value of f(x), so f is not one-to-one.

We can however restrict the domain over which f is defined to extract an invertible function. One such choice would be to restrict to the interval x ≥ 4, which I'll demonstrate below.

Recall the definition of function inverse:

f(f^(-1)(x)) = x

Then for this function, we have

3 (f^(-1)(x) - 4)² + 5 = x

3 (f^(-1)(x) - 4)² = x - 5

(f^(-1)(x) - 4)² = (x - 5)/3

√((f^(-1)(x) - 4)²)= √((x - 5)/3)

For all x, √(x²) = |x|. Then √((f^(-1)(x) - 4)²) = |f^(-1)(x) - 4|, but by restricting x ≥ 4, or x - 4 ≥ 0, we have the condition that f^(-1)(x) - 4 ≥ 0, and so by definition (of the absolute value function) the absolute value reduces to the positive case. In short, we end up with

f^(-1)(x) - 4 = √((x - 5)/3)

and hence the inverse

f^(-1)(x) = √((x - 5)/3) + 4